Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{2p^2 + 18p - 20}{-9p^3 - 45p^2 + 450p}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {2(p^2 + 9p - 10)} {-9p(p^2 + 5p - 50)} $ $ r = -\dfrac{2}{9p} \cdot \dfrac{p^2 + 9p - 10}{p^2 + 5p - 50} $ Next factor the numerator and denominator. $ r = - \dfrac{2}{9p} \cdot \dfrac{(p + 10)(p - 1)}{(p + 10)(p - 5)}$ Assuming $p \neq -10$ , we can cancel the $p + 10$ $ r = - \dfrac{2}{9p} \cdot \dfrac{p - 1}{p - 5}$ Therefore: $ r = \dfrac{ -2(p - 1)}{ 9p(p - 5)}$, $p \neq -10$